Financial/Actuarial Mathematics Seminar

Recent interest in the topic of ``investment with model ambiguity" in the finance, economics and decision theory communities has been motivated largely by efforts to incorporate ``ambiguity aversion", as suggested by experiments such as the Ellsberg Paradox, in the analysis of agent behavior. Closely related work on ``robust portfolio selection" in the optimization community has been driven by the observation that the solutions of classical optimal portfolio selection problems (such as ``mean-variance optimization") are sensitive to statistical errors that can arise during calibration, and that the ``real world" performance of such portfolios can be poor if these errors are ignored. The commonly used method for addressing these issues is some sort of ``worst case" optimization which has led in turn to methodologies such as ``worst case mean-variance" and ``worst case utility maximization". While the ``worst case approach" has its axiomatic foundations in the work of Gilboa and Schmeidler, it has also been criticized for being ``overly pessimistic".

In this talk, we propose and analyze an alternative measure of` ``robust performance". This alternative measure differs from the typical ``worst case expected utility" and ``worst case mean-variance" formulations in that the ``robust performance" of a (dynamic) portfolio is evaluated not only on the basis of its performance when there is an adversarial opponent
(``nature"), but also by its performance relative to a fully informed ``benchmark investor" who behaves optimally given complete knowledge of the otherwise ambiguous model. This ``relative performance" approach has several important properties: (i) decisions arising from this approach are less pessimistic than the portfolios obtained from the typical ``worst case expected utility" and ``worst case mean-variance" formulations, (ii) the dynamic ``relative performance" problem reduces to a convex static optimization problem under reasonable choices of the benchmark portfolio, and (iii) the solution of the ``relative performance" problem coincides with that of a ``Bayesian" portfolio choice problem with an appropriately chosen prior. The static problem is interesting in its own right: it can be interpreted as a less pessimistic alternative to the single period ``worst case mean-variance" problem.

Joint work with J. George Shanthikumar and Thaisiri Watewai